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Means and averaging in the group of rotations. (English) Zbl 1028.47014

Let \(M(3)\) be the algebra of \(3\times 3\) real matrices. The group of rotations in \(\mathbb{R}^3\), denoted by \(SO(3)\), is the Lie group of special rotations in \(\mathbb{R}^3\), defined by \(SO(3)=\{R\in M(3):R^TR=I\) and \(\det R=1\}\).
In this paper, the author gives precise definitions of different, properly invariant notions of mean rotation. Each mean is associated with a metric in \(SO(3)\). The metric induced from the Frobenius inner product gives rise to a mean rotation that is given by the closest special orthogonal matrix to the usual arithmetic mean of the given rotation matrices.
The mean rotation associated with the intrinsic metric on \(SO(3)\) is the Riemannian center of mass of the given rotation matrices. The author shows that the Riemannian mean rotation shares many common features with the geometric mean of positive numbers and the geometric mean of positive Hermitian operators.

MSC:

47A64 Operator means involving linear operators, shorted linear operators, etc.
65F30 Other matrix algorithms (MSC2010)
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